Purely out of curiosity, I decided to mark your answers. You scored 53/75, which equates to an A. The boundary for reaching A was 51, and you needed 61 for an A*. I think an A is very reasonable given that you were going slowly to explain your working, and that you didn't use a calculator or a formula sheet. I'd love to see you try 2023's papers, which will be publically available soon, since they were a lot harder, and absolutely slaughtered me in the my mocks. (mild exaggeration). Hoping that 2024's papers treat me better in the real thing!!!! Score breakdown by question: Q1: 6/6 Q2: 4/4 Q3a: 1/3 Q3b: 3/3 Q4a: 3/4 Q4b: 3/3 Q5a: 2/2 Q5b: 1/4 Q6: 7/7 Q7a: 2/2 Q7b: 4/5 Q8a: 5/5 Q8b: 1/1 Q8c: 4/5 (would've been 5/5 if you used a calculator) Q8d: 0/1 (not answered) Q9: 6/6 Q10: 0/14 (not answered) (if you end up trying 2023's papers, use a calculator and a formula booklet!! they make things SO much easier)
@@jadedtrekkie The answer was wrong. His method would have been correct. He copied the first two rows and columns correctly, but for the third step, he should have copied a, 2, 2 and 3 (the "top left" matrix), not 3, 0, a, and 2 (the "bottom right" matrix). So, the cofactor matrix had an error there.
BPRP, I have watched your videos for years, and in 15 minutes I will take my last calculus exam (calc III at university). I am not worried bc your videos have turned me into a calculus weapon and I only need a 53% for an A anyways. I hope to get an ace anyways and make you proud. Thank you BPRP! Just 2 and a half hours and WE ARE DONE!
@@shishi2177 ok so the exams for the class are proctored and on canvas. Questions are drawn semi-randomly from a large question bank. I think I got really unlucky because I had a ton of questions asking me to integrate over really strange boundaries so I had to improvise a little bit. I still did good tho and at the very worst got like a 60% which keeps me at an A so beyond that I really don't care lol. EDIT: we got an 82% lol let's goooo
It's worth noting that you would have access to a fairly extensive equation booklet in the actual exam - not finished the entire video yet so I might have missed you using it. I also wish last year's papers were released by now... would have been interesting to see you do the paper I had to.
2023 exam series are likely coming out at the start of next week or the end of this week, it's usually just to prevent students cheating in their mocks
For 1b, you can just take the sum of the roots, 2+3i+2-3i+Z = -b/a where a is the coefficient of z^3, b is the coefficient of z^2, so 4+Z=0 and therefore Z = -4 to save a lot of time Similarly, you can calculate the coefficient of z by doing the sum of the product pairs = c/a where c is the coefficient of z and a is the coefficient of z^3 For 8a, the typical technique is to let z=e^ix so 2Cos(nx) = e^inx + e-inx letting n=1 and exponentiating both sides to 6, we get 64Cos^6x = (e^ix + e^-ix)^6 = (e^i6x + e^-i6x) + 6(e^4x + e^-4x) + 15(e^2x + e^-2x) + 20 = 2Cos6x + 12Cos4x + 30Cos2x + 20 therefore 32Cos^6x = Cos6x + 6Cos4x + 15Cos2x + 10 This exam is all about finding the quickest solution which is why so many find it difficult. Much love for showing this on your channel ❤
i was thinking this lol, i wanted to just consider a^2 and b^2 as terms and solve it like a system of equations, but youll still need the ab = 4sqrt(2) after
@@KewlWIS I don't think so, because we add the two equations to get 2a^2 = 32 and a = ±4, then we sub into one of the given equations to get b = ±sqrt(2).
@@bigbrewer3375 good luck!! last year's set of exams was brutal, particularly the core pure 1, if they gave you them for your mocks then you know what I mean. I can't imagine they'll be that bad again
just had my first year exams that decide my ucas, and we had only a little bit of the further cause we did like all of non further in 6 months and then some further after, but there was a volumes of revolution question that i spent about half an hour trying to fix, but it turned out i was getting it wrong because i got the first part of the question wrong, that sucked lol. i ended up getting 26π/(1+ln3) and the answer was 26π/2ln3 or 13π/ln3 oh m y godddd. i hope ive still got some of the marks from that lol, the paper was good otherwise
While the 2022 papers are generally the easiest, whats really impressive is how you do it with no calculator and fodmula booklet. Students in england can use calculators that solve 4 degree polynomials in the exam
At this point, watching you solve problems is like meditation to me. I just sit back and relax as your break down complex sums into simple solutions. You're an unsung hero. Dare I say, slightly better than the organic chemistry tutor.
For question 7, when I was doing this past paper, I was so happy when I realised that if you add a^2-b^2 with a^2+b^2 you get 2a^2=32 which saved me ALOT of time since we know now that a=+/-4. Solving for b gives b=root(2). And we know ab has to be positive so the solutions are 4+root(2)i and -4-root(2)i.
Thanks for doing this paper :). I missed out on Further Maths A-Level in my youth due to poor life choices... I've wondered how this may have impacted me since, for this particular exam I have covered most of this in subsequent learning but great to see you go through and struggle with it. It'd be good to see one that many students in the UK said was "A nightmare" or words to this effect.
I doubt it would have affected you, further maths is only useful if you want to study maths further. Apart from that, most fields of study don’t require more than an A-level understanding.
For Q7, the polar form would be much simpler. Z = r e^ix Z* = r e^(-ix) 1st equation gives r = root(18) 2nd equation gives e^(2ix) = 7/9 + 4 root(2)/9 Using cos(2x) = 2 root(2) / 9, can easily find sin(x) = plus/minus root((1-cos(2x))/2)
On the second part of question 5,you could find the characteristic polyonym of the matrix.There is a theorem stating that a matrix is a root of its characteristic polyonym so you plug the matrix into the polyonym and set it equal to 0. You move the polyonym matrix/constant to the other side, multiply both sides with M^(-1)(you already prooved that there is an inverse) and you get the matrix M^(-1) in terms of M,which should be a valid answer since M is in terms of a
Tackling an A-Level maths exam for the first time is no small feat, especially in one go! It’s inspiring to see the problem-solving process unfold in real time. I’ve been practicing similar problems, and SolutionInn has been super helpful for finding extra examples and improving my skills.
At 56:09 when trying to solve for a and b, instead of substituting you could do the following: a² - b² = 14 (add 2b² to both sides) a² + b² = 14 + 2b² 18 = 14 + 2b² 4 = 2b² 2 = b² √2 = |b| ab =4√2 |a|=4
Amazing video, but rather than the UK A levels, I would recommend the Singapore A levels paper, as it is modified and is usually more fun to attempt than the UK papers
Crazy to watch you do the paper that I sat almost two years ago now. This was the easiest fm paper in a while, because most questions were fairly standard i.e. they relatively closely matched the sort of questions you would find in textbooks. For example, the trig identity you struggled with is something that a student would be expected to basically know by heart, as it is a question directly pulled from an example in the textbook. Iirc paper 2 was a bit harder. Other years papers, such as 2021, were much harder as they required a lot more 'creative' thinking and had a lot more 'funky' questions that weren't just direct application of the material. However, the grade boundaries are adjusted based on the cohorts performance, so 2022 had what are likely to be the highest grade boundaries (the marks you need to get each grade) that fm will ever see. So a much harder paper does not necassarily disadvantage you as the marks you need to get the grade you want are likely to be much lower. I got an A* in these exams, but I am glad I never have to do A levels again because they're so robotic and all about training yourself to do routine questions quickly aso that you have time to think about the more challenging questions. Whereas, at least on my uni course, the exams are much more based around understanding to the point where extensive question practice beyond the point where you understand the concepts is redundant, as all exam questions expect you to apply your understanding to 'unseen' problems and contexts. While this is obviously much more difficult, it allows you to actually spend time attempting to understand the material rather that training yourself to solve a question as quickly as possible
A nice way to do Q1 is to say that for roots alpha, beta and gamma: c/a (3rd term and 1st term of polynomial) = alpha beta + beta gamma + alpha gamma. This gives you the other root in terms of z, which you can then find by solving b/a = alpha + beta + gamma (2+3i + 2-3i + z = 0, z = -4).
For question 9 part c, you can do it by dividing both sides by cosh(x) and then you will have p = 4tanh(x), the range of tanh(x) is between -1 and 1, since there is a 4 beside it, so it’s -4
For qn 6a, to find B and C we can substitute any value for x because the expression is identical on both sides for all values of x. So to find C we let x=0 to get rid of B and then to find B we can let x=1 after knowing the value of C.
Clever way for Q7 b: Once we learn that ab=4root2 we can say a²+2ab+b²=a²+b²+2(ab)= =18+8root2=18+2root32= =(4+root2)² Hence a+b=4+root2 Now from 14=a²-b²=(a+b)(a-b) We get a-b=4-root2. Now it becomes obvious that z=4+iroot2
For problem 3 you can save a little time. The original form y'(cos x) + y(sin x) = F(x) already looks like the product rule, so you can hope to rewrite it as a derivative of a product and avoid having to calculate the integrating factor. However, it is not quite the product rule, because the sign of the second term is wrong. But the insight is that if something "looks like the product rule with the sign of the second term wrong" that means it is almost the quotient rule. You can therefore rewrite the original equation as (y/(cos x))' * cos(x)^2 = F(x) and from there the algebra is very simple.
Consider doing the IB Maths Analysis and Approaches HL Exam. I suggest a paper 1 (the non-calculator paper) or a paper 3 (a fun, challenging, unseen investigation). I would love seeing you do it :)
In question number 8 you can say that z^n + 1/(z^n) has a minimum value of 2 (using AM-GM inequality) and 2cos(n theta) has a maximum value of 2, so cos (n theta) must be always 1. So 32 cos^6(theta)=32, cos 6(theta)=1, 6cos 4(theta) = 6, 15cos 2(theta) = 15. Hence when we add RHS we get 32 and also we get 32 in LHS. Hence Proved.
A lot of people were glad after this paper, it was easier than many other year's exams. 2023 paper was apparently horrific and who knows what they'll do this year.
i sat 2023. it was mainly 'horrific' due to where the questions were placed in the exam. for instance there was a summation q in the first paper that took everyone by surprise but if u look at it again and think through it logically it isnt too hard. you needed to be in a calm mindset to do it which most people were not because it was placed near the last q at which point most people were low on time.
Yeah, 2022 was the year I sat the actual exam for, some of the practice papers had some much funkier questions in them, I think they balanced it out by having more questions than usual in 2022 and making each one slightly more manageable, in order to test a good range of topics across the board. Not a bad idea, but since everyone found it too easy I guess they had to go back to hard papers the year after lol
Thanks for doing this! I never thought id see you doung an a level further maths test. Id love to see you try one of our further maths modules like further pure
For Q7:b, you forgot to give a negative sign to the real part of the second possible complex number z: the possible pairs of RE and IM are (4, sqrt(2)) and (-4,-sqrt(2)). Thanks for the math lessons!
I highly recommend an A-level futher mechanics exam, which tends to have some tricky questions and a tight timer. In fact, I think to get an A* in most of those exams, you only need half of the maximum score
I did mine a few months ago and yeah, it's definitely the toughest paper from Further Maths. Grade boundary for A was 23/50. I study CIE though so I don't know about other boards.
A^2 + B^2 = 18 A^2 - B^2 = 14 This is how us dummies do it (system addition and subtraction) 2A^2 = 32,: A^2 = 16: A = +/- 4 2B^2 = 4 : B^2 = 2: B = +/- SQRT(2)
Woah I’m doing my alevel further maths exam this summer 2024 and I am going on to study Maths and Statistics at uni too.. May I ask what uni and what career path you went into?
@@hejran8017 In 1964 Sheffield University in UK was unique for Statistics course. Those were very different times, ( under 3% of students went to University) when I left university I was offered jobs by the then 2 largest companies in UK, both major world corporations. I chose logistics ( it was a new discipline but could use my statistics/mathematics to transform their inventory policies and production control across countries. Later I was co founder of a contract pharma company employing 500 people. The subjects are useful for developing analytical skills and spotting numerical anomalies in figure data sets.
In Question 5 at 33:33 you misremembered the trick for the inverse of a 3x3 matrix and got the last corner part wrong, it should have been a copy of the top left corner of the original not the bottom right corner. Given matrix (ABC,DEF,GHI) the 4x4 should be (EFDE,HIGH,BCAB,EFDE).
how things have changed . I sat further maths in 1979 , and the Northern Universities Joint Matriculation paper was a mix of pure and applied . There was a selection of questions , and you needed to answer 7 of them (I think it was 7) Our applied maths teacher had saved all the papers for years , and as we approached our exams , we sat a paper every week until we could almost do them in our sleep
with the first question, you can get the coefficients in terms of the roots, in this case, the polynomial has roots α, β, & γ, from this you know α&β, and you know αβγ= 52, so you can find γ, and with that you know that a = αβ + βγ + γα, this is the quick way of doing it that is taught on further maths
OMG, seeing these questions bring me back memories of me sitting in front of the desk in my room, doing past papers after past papers. Just missed mine by 6 years (Jun 2016)
for 9c to solve it quicker u could've done 4 sinh x / cosh x = p so p = 4 tanh x and because we know the range of tanhx is -1 to 1. The range of p is -4 to 4
Method 1) (- x= 3) equation is given Multiplying both sides by (-1) -1*-x=-1*3 Then x=-3 or Method 2) Let the equation be (- x= 3) If we multiply both sides with "MINUS" sign -(- x)= -(3) Then x= -3. Which one is correct or both methods are correct . Please help
omg this is awesome! I agree with the other comments saying you should try the 2023 further core pure paper, but I personally want you to try the year 2 *further pure maths* paper. (the difference is that FPM is an optional module for further maths students where they teach even MORE advanced mathematics like group theory etc), I think it would be good fun
ooooh, or even the STEP (Sixth Term Examination Paper), which in infamously difficult as the enterance exam for studying maths at Cambridge! All of the A level exams are calculator papers, but the enterance exams are all non-calculator so it might be a bit more your style
this was insane to watch you do without a calculator, you should try a further pure 2 paper. when taking further maths you have to choose 2 option modules, it would be interesting to see you do further pure 1 or further pure 2, which are the highest level of pure
As a US high school student, I did this myself for fun and got a 63/75. someone let me know if that is a good score and if I graded myself well. Q1: -0 Q2: -0 Q3: -0 Q4: -1 (forgot the 35 in front) Q5: -1 (made some sign mistakes during the inverse but had more than 5 correct entries) Q6: -0 Q7: -5 (did part b completely wrong) Q8: -3 (copied x instead of x/4 during substitution from part a, idk how many points that is so I took 3) Q9: -2 (part b2 was completely wrong) Q10: -0 (this was super similar to questions we had in diff eq class, glad I remembered all of that)
1:24:05 What is a infinite interval? Because, if it's a interval with a infinite amount of numbers, then any non-degenerate inverval is a infinite interval :| (degenerate inteval being like: [a , a])
I teach A-Level FM and the 2023 papers were comparatively harder than anything seen since the 2018 format. Looking forward to what the 2024 and 2025 papers come up with for my current cohort...
you should have a formuale sheet it would have helped also you could have used Vieta's formulas for the first questions and u can use the formuale cosh(x) = ln(x+sqrt(x^2-1)) then in the end attach plus minus since cosh x will have two solutions for this problem, also ur allowed to ue a graphical calculatorrr as long as it doesnt have alegraic manipulation
This must be paper 1 - the easy quick questions not requiring too much thought. To give yourself a real taste of A' Level Maths, you'll need to do Pure Maths papers 1 & 2 plus Applied Maths or Statistics papers 1 & 2 - at least that was the level in the 80s. I believe they are much simpler these days due to the University entrance requirement for any subject being GCSE Mathematics and English Language. The hardest A' Levels were the Hong Kong A' Levels back then.
If you can find one, I'd be interested how you get on with a Further Maths Special paper. All A-levels before 1989 (I think this is the year they were abolished) had optional Special papers to allow extra grades to be available for mainly Oxford and Cambridge to use. The entry requirement for Cambridge to do Maths was commonly three grade A's, including Maths, Further Maths and a science, and Distinctions in the Special papers in Maths and Further Maths (70%+). I managed A with Distinction in the Special paper in Maths a year early, A in Further Maths, B in Physics and B in Chemistry, so missed out on Cambridge. These were all three-hour exam papers. Nowadays there are three shorter exams for the Maths A-level so the dears aren't put under too much pressure.
I have no relation whatsover to Maths since I chose Biology but its nice to see a teacher like you taking your time with quesgtions and explaining gthem. Wish you all the best
I’m taking a lvl further maths next year and I’m scared 😂 tbh I think if I knew the notations I would be alright as I got what you have to do on some questions I could do with my current knowledge
Hey BPRP, it would be amazing if you could take a look at the IB Maths AA HL papers- they are notoriously one of the hardest, and there are 3 different exams to choose from compared to A-levels
In question 7 there is a small mistake. You found z=4+sqrt(2)i as a solution. The other solution is z=-4-sqrt(2), with a=-4 and b=-sqrt(2). Its a small detail, but your solution is wrong as you basically say that z is a solution as well as z*, meaning z/z* = z*/z = 7/9 + 4sqrt(2)/9i. This is never true for a complex number z. It would mean that z=z*, which would mean that b=0, thus z is not a complex number anymore. You can work it out yourself but I would say its pretty trivial. Another way to look at it is to consider the condition in the question that a*b=4sqrt(2). So either a and b are both positive or both negative.
I'm a Class 12 student in CBSE, preparing for IIT-JEE. Questions were manageable, and I learned a new determinant-solving trick from a video. But, the language in some questions is tricky, even though I'm comfortable with English.
Our calculus teacher asked us to find out what is inverse of function x c^x where is is constant. But for around two days the problem is unsolvable and sir told that find it without W function. If not possible find it with Lambert W function)
I have a a couple of problems with your solution to problem 3: 1. At 16:39 I am wondering how we can divide by cos(x) since technically cos(x)=0 is a possibility, and therefore we are potentially dividing by zero.This is further proven at 20:57 by the final solution where you are setting cos(x) = 0 and thus arriving at x=arccos(0) = (pi)/2! 2. At 16:50 it should remain as e^(2x) and NOT e^(x). Hence at 20:02 This will change your final solution such that it is y = f(x) = cos(x)[e^(2x) + 2] I don't mean to be difficult it's just that as an EE I was forced to be very thorough with algebra :) Thanks!
I havent fully watched this part however what i can gather. You “can” divide by cosx aslong as you remember to include the fact that it could be 0 at the end (which i believe he did). Conventionally it’s better practise to factorise but sometimes that can get messy and sometimes it’s just nicer to simplify the equation by dividing the cosx and dealing with it later. As for the e^2x yeah i think he just forgot the 2. We’ve all done pretty stupid mistakes before lol.
@@JamJam-zy9uw I get what you're saying but even IF he discloses that cos(x) = 0 that's still division by zero because he is assuming cos(x) = 0 thus implying that x=pi/2. In other words, it's still not technically valid, unless I am missing something
@@PowerShellWizard factorising would still assume that cosx=0 at the end of the solution. Dividing by cosx is bad practise however the “bad thing” about dividing by 0 here is you can potentially miss a solution miss such as in (x-2)^2 = x-2 if I divide both sides by x-2 I get x-2=1 hence x = 3 however obviously I’ve missed the other solution when x = 2. Now I could remember that I divided by x-2 and set it equal to 0 to account for that and x-2= 0 therefore x=2 or I could stick with the original equation subtract from both sides to get (x-2)^2-(x-2)=0 factorise (x-2)(x-2-1)=0 and (x-2)(x-3) so either x = 2 or x= 3 which is the I agree the more principled way however both ways do arrive to the same conclusion
Hi BPRP! If you'd like to do another video similar to this, it would be cool if you tried to do one of the Australian VCE Specialist maths exams from last year without knowing the answer guide (exam 1 is technology-free). Thanks for the video :)
I just did my Further Maths Pure exam today… pretty tough but I liked the challenge and managed to answer most questions. It’s slightly different to your one in that mine tests all the content at once in a 2 h 40min sitting… I’m sure you would have loved to do some of the questions though :3
In the third problem it was 2e not e In the next problem it was ln(3) not ln(2) so it should have been n (n+1)/ ln 3 But in the final problem the 3’s cancel.
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Correction to Q5: ruclips.net/video/pXdEBA6xI1c/видео.html
NICE!
So did you graduate? lol
Purely out of curiosity, I decided to mark your answers. You scored 53/75, which equates to an A. The boundary for reaching A was 51, and you needed 61 for an A*. I think an A is very reasonable given that you were going slowly to explain your working, and that you didn't use a calculator or a formula sheet. I'd love to see you try 2023's papers, which will be publically available soon, since they were a lot harder, and absolutely slaughtered me in the my mocks. (mild exaggeration). Hoping that 2024's papers treat me better in the real thing!!!!
Score breakdown by question:
Q1: 6/6
Q2: 4/4
Q3a: 1/3
Q3b: 3/3
Q4a: 3/4
Q4b: 3/3
Q5a: 2/2
Q5b: 1/4
Q6: 7/7
Q7a: 2/2
Q7b: 4/5
Q8a: 5/5
Q8b: 1/1
Q8c: 4/5 (would've been 5/5 if you used a calculator)
Q8d: 0/1 (not answered)
Q9: 6/6
Q10: 0/14 (not answered)
(if you end up trying 2023's papers, use a calculator and a formula booklet!! they make things SO much easier)
This is awesome! Thanks!!
Did he get 1/4 on 5b just because he used an unorthodox method or was the answer actually wrong?
Why was a mark dropped on 7b? Looked fine to me
@@jadedtrekkie The answer was wrong. His method would have been correct. He copied the first two rows and columns correctly, but for the third step, he should have copied a, 2, 2 and 3 (the "top left" matrix), not 3, 0, a, and 2 (the "bottom right" matrix). So, the cofactor matrix had an error there.
@@maxhenderson1890 I thought it was because his second answer should've been -4-sqrt(2)I, but he forgot the negative at the front.
BPRP, I have watched your videos for years, and in 15 minutes I will take my last calculus exam (calc III at university). I am not worried bc your videos have turned me into a calculus weapon and I only need a 53% for an A anyways. I hope to get an ace anyways and make you proud. Thank you BPRP! Just 2 and a half hours and WE ARE DONE!
Wow that’s awesome! Thanks for the message and best wishes to you!
Good luck o7
tell us how it went!
@@shishi2177 ok so the exams for the class are proctored and on canvas. Questions are drawn semi-randomly from a large question bank. I think I got really unlucky because I had a ton of questions asking me to integrate over really strange boundaries so I had to improvise a little bit. I still did good tho and at the very worst got like a 60% which keeps me at an A so beyond that I really don't care lol.
EDIT: we got an 82% lol let's goooo
@@quercus_opuntia goat
oh my gosh! its so weird to see you do my a level further maths paper
i thought these questions were familiar!
What'd you score?
It's worth noting that you would have access to a fairly extensive equation booklet in the actual exam - not finished the entire video yet so I might have missed you using it. I also wish last year's papers were released by now... would have been interesting to see you do the paper I had to.
2023 exam series are likely coming out at the start of next week or the end of this week, it's usually just to prevent students cheating in their mocks
@@AbbieIsQueendo you know if last years gcse papers come out becore i sit them this year?
@@freez.mp4 Yeh they will come out soon, you can ask your teachers for a copy of the papers if you haven't seem them yet and they will usually oblige
@freez.mp4 my teachers have given them out since last week so yea they should be available to teachers
@@AbbieIsQueen I sat one of the 2023 papers in class last week, colleges get them early
For 1b, you can just take the sum of the roots, 2+3i+2-3i+Z = -b/a where a is the coefficient of z^3, b is the coefficient of z^2, so 4+Z=0 and therefore Z = -4 to save a lot of time
Similarly, you can calculate the coefficient of z by doing the sum of the product pairs = c/a where c is the coefficient of z and a is the coefficient of z^3
For 8a, the typical technique is to let z=e^ix so 2Cos(nx) = e^inx + e-inx
letting n=1 and exponentiating both sides to 6, we get
64Cos^6x = (e^ix + e^-ix)^6
= (e^i6x + e^-i6x) + 6(e^4x + e^-4x) + 15(e^2x + e^-2x) + 20
= 2Cos6x + 12Cos4x + 30Cos2x + 20
therefore 32Cos^6x = Cos6x + 6Cos4x + 15Cos2x + 10
This exam is all about finding the quickest solution which is why so many find it difficult.
Much love for showing this on your channel ❤
For 9C you can write 4sinhx = pcoshx as p = 4tanhx
Since the range of 4tanhx is between -4 and 4, the range of values of p is the same -4
Ah!!!! I didn’t think of that!
In part (b), of Q7, use the previous result a^2 + b^2 = 18 with a^2 - b^2 = 14 to get a and b.
i was thinking this lol, i wanted to just consider a^2 and b^2 as terms and solve it like a system of equations, but youll still need the ab = 4sqrt(2) after
@@KewlWIS I don't think so, because we add the two equations to get 2a^2 = 32 and a = ±4, then we sub into one of the given equations to get b = ±sqrt(2).
@@phoquenahol7245 yea but how would you know which pairs to pick? only half of the solutions u gave work
4,-√2 and -4,√2 dont work
Amazing video!! Very much enjoyed it!!
yours too.
I sat the further maths exam last year, it's really exciting to see this video from you lol
I'm sitting my papers this year so this video came out at just the right time to help me with revision
@@bigbrewer3375 good luck!! last year's set of exams was brutal, particularly the core pure 1, if they gave you them for your mocks then you know what I mean. I can't imagine they'll be that bad again
just had my first year exams that decide my ucas, and we had only a little bit of the further cause we did like all of non further in 6 months and then some further after, but there was a volumes of revolution question that i spent about half an hour trying to fix, but it turned out i was getting it wrong because i got the first part of the question wrong, that sucked lol. i ended up getting 26π/(1+ln3) and the answer was 26π/2ln3 or 13π/ln3 oh m y godddd. i hope ive still got some of the marks from that lol, the paper was good otherwise
@@abbyskywind Thanks! fingers crossed i'll do well
@@abbyskywindomg if that volumes of revolution question was from last years core pure 1 i s2g
While the 2022 papers are generally the easiest, whats really impressive is how you do it with no calculator and fodmula booklet. Students in england can use calculators that solve 4 degree polynomials in the exam
This on top of the fact youre explaining it while doing it is probably why you struggled with timing, this is a really impressive video
wasn't that because covid hit the worse ?
@@tar244you8 There are no non-calc papers at a level.
You can use the calculators yes, but you won’t get the marks as you need to show workings.
At this point, watching you solve problems is like meditation to me. I just sit back and relax as your break down complex sums into simple solutions. You're an unsung hero. Dare I say, slightly better than the organic chemistry tutor.
LMAOOOO😭
For question 7, when I was doing this past paper, I was so happy when I realised that if you add a^2-b^2 with a^2+b^2 you get 2a^2=32 which saved me ALOT of time since we know now that a=+/-4. Solving for b gives b=root(2). And we know ab has to be positive so the solutions are 4+root(2)i and -4-root(2)i.
Going in raw no calculator, no formulae is honestly the kind of thing I have nightmares about so this is pretty crazy.
Thanks for doing this paper :). I missed out on Further Maths A-Level in my youth due to poor life choices... I've wondered how this may have impacted me since, for this particular exam I have covered most of this in subsequent learning but great to see you go through and struggle with it. It'd be good to see one that many students in the UK said was "A nightmare" or words to this effect.
I doubt it would have affected you, further maths is only useful if you want to study maths further.
Apart from that, most fields of study don’t require more than an A-level understanding.
16:50 You did a mistake there is e^2x not e^x.
@ PS_MEMES.XC2 -- You made an error. It is e^(2x), because the exponent must be inside grouping symbols because of the Order of Operations.
@@forcelifeforceshut up
@@forcelifeforce you know what they meant
At 59:03 the next value of z will be -4-sqrt(2). because product of a and b must be positive. as ab=4 sqrt(2).
For Q7, the polar form would be much simpler.
Z = r e^ix
Z* = r e^(-ix)
1st equation gives r = root(18)
2nd equation gives e^(2ix) = 7/9 + 4 root(2)/9
Using cos(2x) = 2 root(2) / 9, can easily find sin(x) = plus/minus root((1-cos(2x))/2)
On the second part of question 5,you could find the characteristic polyonym of the matrix.There is a theorem stating that a matrix is a root of its characteristic polyonym so you plug the matrix into the polyonym and set it equal to 0. You move the polyonym matrix/constant to the other side, multiply both sides with M^(-1)(you already prooved that there is an inverse) and you get the matrix M^(-1) in terms of M,which should be a valid answer since M is in terms of a
That is a great trick
Unfortunately for the students sitting this paper the Cayley-Hamilton isn't part of the module - it is a nice trick though!
@@olivermaclean8564it is if they did the FP2 module, this is CP1, a mandatory module. Some students will know it who took FP2
Tackling an A-Level maths exam for the first time is no small feat, especially in one go! It’s inspiring to see the problem-solving process unfold in real time. I’ve been practicing similar problems, and SolutionInn has been super helpful for finding extra examples and improving my skills.
At 56:09 when trying to solve for a and b, instead of substituting you could do the following:
a² - b² = 14 (add 2b² to both sides)
a² + b² = 14 + 2b²
18 = 14 + 2b²
4 = 2b²
2 = b²
√2 = |b|
ab =4√2
|a|=4
Amazing video, but rather than the UK A levels, I would recommend the Singapore A levels paper, as it is modified and is usually more fun to attempt than the UK papers
Crazy to watch you do the paper that I sat almost two years ago now. This was the easiest fm paper in a while, because most questions were fairly standard i.e. they relatively closely matched the sort of questions you would find in textbooks. For example, the trig identity you struggled with is something that a student would be expected to basically know by heart, as it is a question directly pulled from an example in the textbook. Iirc paper 2 was a bit harder. Other years papers, such as 2021, were much harder as they required a lot more 'creative' thinking and had a lot more 'funky' questions that weren't just direct application of the material. However, the grade boundaries are adjusted based on the cohorts performance, so 2022 had what are likely to be the highest grade boundaries (the marks you need to get each grade) that fm will ever see. So a much harder paper does not necassarily disadvantage you as the marks you need to get the grade you want are likely to be much lower. I got an A* in these exams, but I am glad I never have to do A levels again because they're so robotic and all about training yourself to do routine questions quickly aso that you have time to think about the more challenging questions. Whereas, at least on my uni course, the exams are much more based around understanding to the point where extensive question practice beyond the point where you understand the concepts is redundant, as all exam questions expect you to apply your understanding to 'unseen' problems and contexts. While this is obviously much more difficult, it allows you to actually spend time attempting to understand the material rather that training yourself to solve a question as quickly as possible
I just sat the 2024 CP1 paper and it was easier than this lmao
Which course are u doing
@@ggcvbvnbcby1424 aeronautical engineering at Imperial
A nice way to do Q1 is to say that for roots alpha, beta and gamma: c/a (3rd term and 1st term of polynomial) = alpha beta + beta gamma + alpha gamma. This gives you the other root in terms of z, which you can then find by solving b/a = alpha + beta + gamma (2+3i + 2-3i + z = 0, z = -4).
I love this vid. I'm not good with math, but i love it. I find it fascinating to solve things and search for limits or lack there of.
For question 9 part c, you can do it by dividing both sides by cosh(x) and then you will have p = 4tanh(x), the range of tanh(x) is between -1 and 1, since there is a 4 beside it, so it’s -4
For qn 6a, to find B and C we can substitute any value for x because the expression is identical on both sides for all values of x. So to find C we let x=0 to get rid of B and then to find B we can let x=1 after knowing the value of C.
On #7, the second answer should be -4 - (root 2)i
Clever way for Q7 b:
Once we learn that ab=4root2
we can say a²+2ab+b²=a²+b²+2(ab)=
=18+8root2=18+2root32=
=(4+root2)²
Hence a+b=4+root2
Now from 14=a²-b²=(a+b)(a-b)
We get a-b=4-root2.
Now it becomes obvious that
z=4+iroot2
For problem 3 you can save a little time. The original form y'(cos x) + y(sin x) = F(x) already looks like the product rule, so you can hope to rewrite it as a derivative of a product and avoid having to calculate the integrating factor. However, it is not quite the product rule, because the sign of the second term is wrong. But the insight is that if something "looks like the product rule with the sign of the second term wrong" that means it is almost the quotient rule.
You can therefore rewrite the original equation as (y/(cos x))' * cos(x)^2 = F(x) and from there the algebra is very simple.
it is the quotient rule, though. Bring the cos^2(x) term across then (cosx.dy/dx+y.sinx)/cos^2(x)= d/dy (y/cosx). Should be e^2x throughout.
Consider doing the IB Maths Analysis and Approaches HL Exam. I suggest a paper 1 (the non-calculator paper) or a paper 3 (a fun, challenging, unseen investigation). I would love seeing you do it :)
In question number 8 you can say that z^n + 1/(z^n) has a minimum value of 2 (using AM-GM inequality) and 2cos(n theta) has a maximum value of 2, so cos (n theta) must be always 1.
So 32 cos^6(theta)=32, cos 6(theta)=1, 6cos 4(theta) = 6, 15cos 2(theta) = 15. Hence when we add RHS we get 32 and also we get 32 in LHS.
Hence Proved.
there are also option modules for further maths, and the "highest" level pure work for a levels is on the further pure 2 paper. Please try that
A lot of people were glad after this paper, it was easier than many other year's exams. 2023 paper was apparently horrific and who knows what they'll do this year.
i sat 2023. it was mainly 'horrific' due to where the questions were placed in the exam. for instance there was a summation q in the first paper that took everyone by surprise but if u look at it again and think through it logically it isnt too hard. you needed to be in a calm mindset to do it which most people were not because it was placed near the last q at which point most people were low on time.
Yeah, 2022 was the year I sat the actual exam for, some of the practice papers had some much funkier questions in them, I think they balanced it out by having more questions than usual in 2022 and making each one slightly more manageable, in order to test a good range of topics across the board. Not a bad idea, but since everyone found it too easy I guess they had to go back to hard papers the year after lol
Yeah, I'm scared. Core pure 1 is in 23 days now.
@@Zephyr_197 good luck!, I'm doing regular maths y12 and Further in y13 so I'll probably end up doing your paper in a mock lol
@@Zephyr_19720 days now 💀
was so so awesome to see you doing my a level paper, now 2 years into uni :D
Thanks for doing this! I never thought id see you doung an a level further maths test. Id love to see you try one of our further maths modules like further pure
For Q7:b, you forgot to give a negative sign to the real part of the second possible complex number z: the possible pairs of RE and IM are (4, sqrt(2)) and (-4,-sqrt(2)). Thanks for the math lessons!
I highly recommend an A-level futher mechanics exam, which tends to have some tricky questions and a tight timer. In fact, I think to get an A* in most of those exams, you only need half of the maximum score
I did mine a few months ago and yeah, it's definitely the toughest paper from Further Maths. Grade boundary for A was 23/50. I study CIE though so I don't know about other boards.
I don't even take A level maths why am I watching this
A^2 + B^2 = 18
A^2 - B^2 = 14
This is how us dummies do it (system addition and subtraction)
2A^2 = 32,: A^2 = 16: A = +/- 4
2B^2 = 4 : B^2 = 2: B = +/- SQRT(2)
Finally, someone who takes on the A2 Further Mathematics paper. Many of the other videos on YT chicken-out and only go as far as the AS paper.
Fascinating. I passed A level Further maths in 1964....with a B grade. Did Maths and Statistics at Uni.
Woah I’m doing my alevel further maths exam this summer 2024 and I am going on to study Maths and Statistics at uni too.. May I ask what uni and what career path you went into?
@@hejran8017 In 1964 Sheffield University in UK was unique for Statistics course. Those were very different times, ( under 3% of students went to University) when I left university I was offered jobs by the then 2 largest companies in UK, both major world corporations. I chose logistics ( it was a new discipline but could use my statistics/mathematics to transform their inventory policies and production control across countries. Later I was co founder of a contract pharma company employing 500 people. The subjects are useful for developing analytical skills and spotting numerical anomalies in figure data sets.
In Question 5 at 33:33 you misremembered the trick for the inverse of a 3x3 matrix and got the last corner part wrong, it should have been a copy of the top left corner of the original not the bottom right corner. Given matrix (ABC,DEF,GHI) the 4x4 should be (EFDE,HIGH,BCAB,EFDE).
this guy is going to solve all the math paper around the world
Q8 abs(z^n+1/z^n)>=2 abs(2cosn Teta.)
how things have changed . I sat further maths in 1979 , and the Northern Universities Joint Matriculation paper was a mix of pure and applied . There was a selection of questions , and you needed to answer 7 of them (I think it was 7) Our applied maths teacher had saved all the papers for years , and as we approached our exams , we sat a paper every week until we could almost do them in our sleep
Are you saying exams have gotten easier?
@@MVG7 not at all . just a different style
@@ae112 fair enough
with the first question, you can get the coefficients in terms of the roots, in this case, the polynomial has roots α, β, & γ, from this you know α&β, and you know αβγ= 52, so you can find γ, and with that you know that a = αβ + βγ + γα, this is the quick way of doing it that is taught on further maths
my bad, αβγ = -52
8. (d) The equation of the curve may not be suitable.
Or, the measurements may not be accurate.
Or, the paperweight may not be smooth.
OMG, seeing these questions bring me back memories of me sitting in front of the desk in my room, doing past papers after past papers. Just missed mine by 6 years (Jun 2016)
for 9c to solve it quicker u could've done 4 sinh x / cosh x = p so p = 4 tanh x and because we know the range of tanhx is -1 to 1. The range of p is -4 to 4
I really like his passion for mathematics.
I love that solving some test on RUclips can be done "for entertaining purposes" 😁
You don’t need to take 1/2 the volume in equation number 8, just integrate from 0 to 4.
Sat this paper last year, now im doing maths at uni :))
Method 1)
(- x= 3) equation is given
Multiplying both sides by (-1)
-1*-x=-1*3
Then x=-3
or
Method 2)
Let the equation be (- x= 3)
If we multiply both sides with "MINUS" sign
-(- x)= -(3)
Then x= -3.
Which one is correct or both methods are correct .
Please help
16:48 when he cancels the cosines, he forgot that e was raised to 2x or I’m wrong?
omg this is awesome! I agree with the other comments saying you should try the 2023 further core pure paper, but I personally want you to try the year 2 *further pure maths* paper. (the difference is that FPM is an optional module for further maths students where they teach even MORE advanced mathematics like group theory etc), I think it would be good fun
ooooh, or even the STEP (Sixth Term Examination Paper), which in infamously difficult as the enterance exam for studying maths at Cambridge! All of the A level exams are calculator papers, but the enterance exams are all non-calculator so it might be a bit more your style
@@richtigmann1 he has already covered step questions before ruclips.net/video/05GQzRfW26c/видео.html
this was insane to watch you do without a calculator, you should try a further pure 2 paper. when taking further maths you have to choose 2 option modules, it would be interesting to see you do further pure 1 or further pure 2, which are the highest level of pure
As a US high school student, I did this myself for fun and got a 63/75. someone let me know if that is a good score and if I graded myself well.
Q1: -0
Q2: -0
Q3: -0
Q4: -1 (forgot the 35 in front)
Q5: -1 (made some sign mistakes during the inverse but had more than 5 correct entries)
Q6: -0
Q7: -5 (did part b completely wrong)
Q8: -3 (copied x instead of x/4 during substitution from part a, idk how many points that is so I took 3)
Q9: -2 (part b2 was completely wrong)
Q10: -0 (this was super similar to questions we had in diff eq class, glad I remembered all of that)
That would be an A* well done!
What highschool offers differentials as a class😭.
My highschool didn't even have calculus, so I had to self study it.
@@abel3557 my high school doesn't offer it, I took the class as well as linear algebra at a local 4-year university.
1:24:05 What is a infinite interval? Because, if it's a interval with a infinite amount of numbers, then any non-degenerate inverval is a infinite interval :| (degenerate inteval being like: [a , a])
An infinite interval is an unbounded interval
On question 3 you wrote e^x instead of e^2x. Is it that right?. The answer would be Ans: y = cos(x) * (1/2 * e^(2x) + 5/2). I like your videos.
Thanks for this. In question 3, why did e^(2x) become e^x?
Error.
Absolutely tragic
I teach A-Level FM and the 2023 papers were comparatively harder than anything seen since the 2018 format.
Looking forward to what the 2024 and 2025 papers come up with for my current cohort...
Love your work! For Q3 I can't help but notice in the second line you accidently wrote e^x instead of e^2x? I got y=1/2*cosx(e^2x+5) for my answer
you should have a formuale sheet it would have helped also you could have used Vieta's formulas for the first questions and u can use the formuale cosh(x) = ln(x+sqrt(x^2-1)) then in the end attach plus minus since cosh x will have two solutions for this problem, also ur allowed to ue a graphical calculatorrr as long as it doesnt have alegraic manipulation
This must be paper 1 - the easy quick questions not requiring too much thought. To give yourself a real taste of A' Level Maths, you'll need to do Pure Maths papers 1 & 2 plus Applied Maths or Statistics papers 1 & 2 - at least that was the level in the 80s. I believe they are much simpler these days due to the University entrance requirement for any subject being GCSE Mathematics and English Language. The hardest A' Levels were the Hong Kong A' Levels back then.
This brings back memories of just about passing further maths in 1977. Can't remember much about that paper!
this is a great solution thank you
If you can find one, I'd be interested how you get on with a Further Maths Special paper. All A-levels before 1989 (I think this is the year they were abolished) had optional Special papers to allow extra grades to be available for mainly Oxford and Cambridge to use. The entry requirement for Cambridge to do Maths was commonly three grade A's, including Maths, Further Maths and a science, and Distinctions in the Special papers in Maths and Further Maths (70%+). I managed A with Distinction in the Special paper in Maths a year early, A in Further Maths, B in Physics and B in Chemistry, so missed out on Cambridge. These were all three-hour exam papers. Nowadays there are three shorter exams for the Maths A-level so the dears aren't put under too much pressure.
I have no relation whatsover to Maths since I chose Biology but its nice to see a teacher like you taking your time with quesgtions and explaining gthem. Wish you all the best
I’m taking a lvl further maths next year and I’m scared 😂 tbh I think if I knew the notations I would be alright as I got what you have to do on some questions I could do with my current knowledge
UK maths student gang is very happy :)
Thanks You Professor Sir! I am a 65 Years Old ,I am from ROMANIA and I like Calculus invented by the genius polymath LEIBNIZ!
Hey BPRP, it would be amazing if you could take a look at the IB Maths AA HL papers- they are notoriously one of the hardest, and there are 3 different exams to choose from compared to A-levels
You might also want to check IB Math AA HL
36:00 Inverse Matrix Shortcut
@38:43 Why "-3a -14" and not "-3a -4" ?
In question 7 there is a small mistake. You found z=4+sqrt(2)i as a solution. The other solution is z=-4-sqrt(2), with a=-4 and b=-sqrt(2).
Its a small detail, but your solution is wrong as you basically say that z is a solution as well as z*, meaning z/z* = z*/z = 7/9 + 4sqrt(2)/9i. This is never true for a complex number z. It would mean that z=z*, which would mean that b=0, thus z is not a complex number anymore. You can work it out yourself but I would say its pretty trivial.
Another way to look at it is to consider the condition in the question that a*b=4sqrt(2). So either a and b are both positive or both negative.
I'm a Class 12 student in CBSE, preparing for IIT-JEE. Questions were manageable, and I learned a new determinant-solving trick from a video. But, the language in some questions is tricky, even though I'm comfortable with English.
Our calculus teacher asked us to find out what is inverse of function x c^x where is is constant. But for around two days the problem is unsolvable and sir told that find it without W function. If not possible find it with Lambert W function)
In the 7th problem, i think you should write like: a⁴+14a²-32=0 but you wrote -14. But your did awesome. Good like
The amount of times he sighed in this video. I feel his pain.
Please do an IB Mathematics AA HL paper 1 next!
or PAPER 3
that's a lot of markers in the back
5: you did mistake, -3a-4 not -3a-14
I have a a couple of problems with your solution to problem 3:
1. At 16:39 I am wondering how we can divide by cos(x) since technically cos(x)=0 is a possibility, and therefore we are potentially dividing by zero.This is further proven at 20:57 by the final solution where you are setting cos(x) = 0 and thus arriving at x=arccos(0) = (pi)/2!
2. At 16:50 it should remain as e^(2x) and NOT e^(x). Hence at 20:02 This will change your final solution such that it is y = f(x) = cos(x)[e^(2x) + 2]
I don't mean to be difficult it's just that as an EE I was forced to be very thorough with algebra :)
Thanks!
I havent fully watched this part however what i can gather. You “can” divide by cosx aslong as you remember to include the fact that it could be 0 at the end (which i believe he did). Conventionally it’s better practise to factorise but sometimes that can get messy and sometimes it’s just nicer to simplify the equation by dividing the cosx and dealing with it later. As for the e^2x yeah i think he just forgot the 2. We’ve all done pretty stupid mistakes before lol.
@@JamJam-zy9uw I get what you're saying but even IF he discloses that cos(x) = 0 that's still division by zero because he is assuming cos(x) = 0 thus implying that x=pi/2. In other words, it's still not technically valid, unless I am missing something
@@PowerShellWizard factorising would still assume that cosx=0 at the end of the solution. Dividing by cosx is bad practise however the “bad thing” about dividing by 0 here is you can potentially miss a solution miss such as in (x-2)^2 = x-2 if I divide both sides by x-2 I get x-2=1 hence x = 3 however obviously I’ve missed the other solution when x = 2. Now I could remember that I divided by x-2 and set it equal to 0 to account for that and x-2= 0 therefore x=2 or I could stick with the original equation subtract from both sides to get (x-2)^2-(x-2)=0 factorise (x-2)(x-2-1)=0 and (x-2)(x-3) so either x = 2 or x= 3 which is the I agree the more principled way however both ways do arrive to the same conclusion
wish me luck on this in 3 weeks 🙏🙏 i need a*.
Solve a^2 + b^2 = 18 and a^2 - b^2 = 14 simultaneously
“What does this have to do with z?!”😂😂😂 that was very funny
1/2 to the 6th power is 64 so 32/64 = 1/2
It would be interesting to hear your opinion on UK maths exams papers in general. GCSE is for 15-16 year olds while A-levels are for 17-18 year olds.
Would love to see you do some of the nzqa scholarship calculus papers. They have some pretty interesting questions spanning a wide range of topics.
Hi BPRP! If you'd like to do another video similar to this, it would be cool if you tried to do one of the Australian VCE Specialist maths exams from last year without knowing the answer guide (exam 1 is technology-free). Thanks for the video :)
I think you need to multiply each cofactor by -1 raised to the sum of row and column
Please part 2 ; I'm from Jordan I'm Arab, I love your work.....
Could you make a video on partial fractions? Your method seems faster than the two methods I was taught
We need more videos about a level maths pls
I just did my Further Maths Pure exam today… pretty tough but I liked the challenge and managed to answer most questions. It’s slightly different to your one in that mine tests all the content at once in a 2 h 40min sitting… I’m sure you would have loved to do some of the questions though :3
which board did you do?
In the third problem it was 2e not e
In the next problem it was ln(3) not ln(2) so it should have been n (n+1)/ ln 3
But in the final problem the 3’s cancel.
I think 59:00 should be -4-i(sqrt(2))
End of number 7 is 4+rt2i or -4-rt2i